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07/07/2023
Fermat Distances: Metric Approximation, Spectral Convergence, and Clustering Algorithms
We analyze the convergence properties of Fermat distances, a family of d...
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07/05/2023
Continuum Limits of Ollivier's Ricci Curvature on data clouds: pointwise consistency and global lower bounds
Let ℳ⊆ℝ^d denote a low-dimensional manifold and let 𝒳= { x_1, …, x_n } b...
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05/30/2023
It begins with a boundary: A geometric view on probabilistically robust learning
Although deep neural networks have achieved super-human performance on m...
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05/04/2023
FedCBO: Reaching Group Consensus in Clustered Federated Learning through Consensus-based Optimization
Federated learning is an important framework in modern machine learning ...
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04/28/2023
On the existence of solutions to adversarial training in multiclass classification
We study three models of the problem of adversarial training in multicla...
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01/09/2023
On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it
We propose iterative algorithms to solve adversarial problems in a varie...
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10/13/2022
Wasserstein Barycenter-based Model Fusion and Linear Mode Connectivity of Neural Networks
Based on the concepts of Wasserstein barycenter (WB) and Gromov-Wasserst...
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09/29/2022
Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective
We study a general matrix optimization problem with a fixed-rank positiv...
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09/06/2022
Rates of Convergence for Regression with the Graph Poly-Laplacian
In the (special) smoothing spline problem one considers a variational pr...
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07/03/2022
Mathematical Foundations of Graph-Based Bayesian Semi-Supervised Learning
In recent decades, science and engineering have been revolutionized by a...
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04/27/2022
The Multimarginal Optimal Transport Formulation of Adversarial Multiclass Classification
We study a family of adversarial multiclass classification problems and ...
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11/26/2021
The Geometry of Adversarial Training in Binary Classification
We establish an equivalence between a family of adversarial training pro...
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09/13/2021
On the regularized risk of distributionally robust learning over deep neural networks
In this paper we explore the relation between distributionally robust le...
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08/18/2021
Clustering dynamics on graphs: from spectral clustering to mean shift through Fokker-Planck interpolation
In this work we build a unifying framework to interpolate between densit...
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07/28/2021
Large sample spectral analysis of graph-based multi-manifold clustering
In this work we study statistical properties of graph-based algorithms f...
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11/21/2020
Adversarial Classification: Necessary conditions and geometric flows
We study a version of adversarial classification where an adversary is e...
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07/13/2020
Lipschitz regularity of graph Laplacians on random data clouds
In this paper we study Lipschitz regularity of elliptic PDEs on geometri...
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06/26/2020
Semi-discrete optimization through semi-discrete optimal transport: a framework for neural architecture search
In this paper we introduce a theoretical framework for semi-discrete opt...
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06/26/2020
Traditional and accelerated gradient descent for neural architecture search
In this paper, we introduce two algorithms for neural architecture searc...
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04/20/2020
From graph cuts to isoperimetric inequalities: Convergence rates of Cheeger cuts on data clouds
In this work we study statistical properties of graph-based clustering a...
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03/18/2020
Data-Driven Forward Discretizations for Bayesian Inversion
This paper suggests a framework for the learning of discretizations of e...
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10/29/2019
Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs
In this paper we improve the spectral convergence rates for graph-based ...
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04/06/2019
Local Regularization of Noisy Point Clouds: Improved Global Geometric Estimates and Data Analysis
Several data analysis techniques employ similarity relationships between...
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01/30/2019
Geometric structure of graph Laplacian embeddings
We analyze the spectral clustering procedure for identifying coarse stru...
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01/29/2019
A maximum principle argument for the uniform convergence of graph Laplacian regressors
We study asymptotic consistency guarantees for a non-parametric regressi...
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01/29/2019
Variational Characterizations of Local Entropy and Heat Regularization in Deep Learning
The aim of this paper is to provide new theoretical and computational un...
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08/09/2018
Spatial extreme values: variational techniques and stochastic integrals
This work employs variational techniques to revisit and expand the const...
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01/30/2018
Error estimates for spectral convergence of the graph Laplacian on random geometric graphs towards the Laplace--Beltrami operator
We study the convergence of the graph Laplacian of a random geometric gr...
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10/20/2017
On the Consistency of Graph-based Bayesian Learning and the Scalability of Sampling Algorithms
A popular approach to semi-supervised learning proceeds by endowing the ...
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06/22/2017
Continuum Limit of Posteriors in Graph Bayesian Inverse Problems
We consider the problem of recovering a function input of a differential...
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02/11/2017
Gromov-Hausdorff limit of Wasserstein spaces on point clouds
We consider a point cloud X_n := { x_1, ..., x_n } uniformly distributed...
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07/03/2016
Variational limits of k-NN graph based functionals on data clouds
We consider i.i.d. samples x_1, ..., x_n from a measure ν with density s...
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07/01/2016
A new analytical approach to consistency and overfitting in regularized empirical risk minimization
This work considers the problem of binary classification: given training...
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08/08/2015
A variational approach to the consistency of spectral clustering
This paper establishes the consistency of spectral approaches to data cl...
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11/24/2014
Consistency of Cheeger and Ratio Graph Cuts
This paper establishes the consistency of a family of graph-cut-based al...
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03/25/2014